Statistics for Linguistics Students

Michaelmas 2004Week 7

Bettina Braunwww.phon.ox.ac.uk/~bettina/teaching.html

Overview

• Problems from last assignment

• Correlation analyses

• Repeated measures ANOVA– One-way (one IV)– Two-way (two IVs)

• Transformations

Chi-square using SPSS

• Organisation of data:

Chi-square using SPSS

• Where to find it…

Chi-square using SPSS

• How to interpret the output

Table similar to ours

Result: sign. interaction (x2=5.7, df=1, p=0.017

More on interactions

North South

MaleFemale

North South

No effect of region, nor gender, no interaction

Effect of region and gender no interaction

North South

No effect of gender, effect of region, no interaction

North South

Effect of region and gender and interaction

North South

Effect of region and gender and interaction

Correlation analyses

• Often found in exploratory research– You do not test the effect of an independent

variable on the dependent one– But see what relationships hold between two

or more variables

Correlation coefficients

• Scatterplots helpful to see whether it is a linear relationship…

r = -1Neg. corr.

r = 0no corr.

r = 1pos. corr.

Bivariate correlation

• Do you expect a correlation between the two variables?

• Try “line-fitting” by eye

?

Pearson correlation

• T-test is used to test if corr. coefficient is different from 0 ( => data must be interval!)

• If not, use Spearmans correlation (non-parametric)

Pearson correlation

• Correlation coefficient– For interval data– For linear relationships

• r2 is the proportion of variation of one variable that is “explained” by the other

• Note: even a highly significant correlation does not imply a causal relationship (e.g. There might be another variable influencing both!)

Repeated measures ANOVA

• Recall:– In between-subjects designs large individual

differences – repeated measures (aka within-subjects) has

all participants in all levels of all conditions

• Problems: – Practice effect (carry-over) effect

Missing data

• You need to have data for every subject in every condition

• If this is not the case, you cannot include this subject

• If your design becomes inbalanced by the exclusion of a subject, you should randomly exclude a subject from the other group as well (or run another subject for the group with the exclusion)

Requirements for repeated measures ANOVA

• Same as for between-subjects ANOVA• You can have within- and between-subject

factors (e.g. boys vs. girls, producing /a/ and /i/ and /u/)

• Covariates– factors that might have an effect on the within-

subjects factor– Note: covariates can also be specified for

between-subjects designs!

Covariates: example

• You want to study French skills when using 2 different text-books. Students are randomly assigned to 2 groups. If you have the IQ of these students, you can decrease the variability within the groups by using IQ as covariate

• Problem: if the covariate is correlated with between-groups factor as well, F-value might get smaller (less significant)!

• You can also assess interaction between covariates and between-groups factors (e.g. one textbook might be better suited for smart students)

One-way repeated measures ANOVA in SPSS

2

3

1. Define new name and levels for within-subject factor

One-way repeated measures ANOVA in SPSS

• Factor-name• Four levels of the

within-subjects variable

• Enter between-subjects and covariates (if applicable)

Post-hoc tests for within-subjects variables

• SPSS does not allow you to do post-hoc tests for within-subjects variables

• Instead do “Contrasts” and define them as“Repeated”

2

1

Post-hoc tests for within-subjects variables

• You can also askfor a comparsonof means

SPSS output: test of Sphericity

• Test for homgeneity of covariances among scores of within-subjecs factors

• Only calculated if variable has more than 2 levels

If test is significant, you have to reject the null-hypothesis that the variances are homogenious

SPSS output: within-subjects contrasts

• Post-hoc test for within-subjects variables

3 x 3 designs

• 3 x 3 between subjects

Factor B (between)

B1 B2 B3

A1 Group1 Group2 Group3

A2 Group4 Group5 Group6

A3 Group7 Group8 Group9

3 x 3 designs

• 3 x 3 within subjects

Factor B (witin)

B1 B2 B3

A1 Group1

A2

A3

Group1 Group1

Group1

Group1

Group1

Group1

Group1

Group1

3 x 3 designs

• 3 x 3 mixed design

Factor B (witin)

B1 B2 B3

Factor A(between)

A1 Group1

A2

A3

Group1 Group1

Group2

Group3

Group2 Group2

Group3 Group3

Data transformation

• If you want to caculate an ANOVA but your interval data is not normally distributed (i.e. skewed) you can use mathematical transformations

• The type of transformation depends on the shape of the sample distribution

• NOTE: – After transforming data, check the resulting

distribution again for normality!– Note that your data becomes ordinal by transforming

it!! (but you can do an ANOVA with it)

What kind of tranformation?

e.g.f(x) = x1.5

e.g.f(x) = log(x)f(x) = atan(x)

Transformation